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On the Dynamics of Bohmian Measures

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Abstract

The present work is devoted to the study of dynamical features of Bohmian measures, recently introduced by the authors. We rigorously prove that for sufficiently smooth wave functions the corresponding Bohmian measure furnishes a distributional solution of a nonlinear Vlasov-type equation. Moreover, we study the associated defect measures appearing in the classical limit. In one space dimension, this yields a new connection between mono-kinetic Wigner and Bohmian measures. In addition, we shall study the dynamics of Bohmian measures associated to so-called semi-classical wave packets. For these type of wave functions, we prove local in-measure convergence of a rescaled sequence of Bohmian trajectories towards the classical Hamiltonian flow on phase space. Finally, we construct an example of wave functions whose limiting Bohmian measure is not mono-kinetic but nevertheless equals the associated Wigner measure.

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Authors and Affiliations

  1. MCSE Division, King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Saudi Arabia

    Peter Markowich

  2. CNRS and CMLS, Ecole Polytechnique 91, 128, Palaiseau cedex, France

    Thierry Paul

  3. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL, 60607, USA

    Christof Sparber

Authors
  1. Peter Markowich
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  2. Thierry Paul
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  3. Christof Sparber
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Correspondence to Christof Sparber.

Additional information

Communicated by A. Mielke

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Markowich, P., Paul, T. & Sparber, C. On the Dynamics of Bohmian Measures. Arch Rational Mech Anal 205, 1031–1054 (2012). https://doi.org/10.1007/s00205-012-0528-1

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  • DOI: https://doi.org/10.1007/s00205-012-0528-1

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